How can I determine the bias and variance of a random forrest?
To gain a good understanding of this, I recommend first reading about the trade-off between bias and variance in ML and AI methods.

A great article on this topic that I recommend as a light mathematical introduction is this:
https://towardsdatascience.com/understanding-the-bias-variance-tradeoff-165e6942b229

In short:
Bias represents the models effort to generalize samples, as opposed to Variance that represents the models effort to conform to new data.
A high bias, low variance model will thus look more like a straight(underfitted) line, while a low bias, high variance model will look jagged and all-over the place(overfitted).

In essence, you need to find a balance between the two to avoid both overfitting(high variance, low bias) and underfitting(high bias, low variance) for your specific application.


  But how can I determine this for a model such as a Random Forrest classifier?


To determine your models bias and variance configuration(if either is too high/low), you can look at the models performance on the validation and test set.
The very reason we divide our data into training-validation-test sets, is so that we can validate the models performance when it is presented with samples it has not seen during training.